## The Bagel Award-Winning Polar Graph Lesson

This week my PreAP Precal classes delved into polar coordinates and graphs.  Last year, we did some converting back and forth between polar and rectangular equations and then spent a day on graphing polar equations using t-tables.  While I did enjoy seeing where the petals and loops came from, we only got through 6 graphs and we didn’t really delve into recognizing the different equations and being able to do a quick sketch.

Then our school decided to teach Calculus BC this year and–oops!–turns out that skill is kind of an important thing.  Plus I’m helping to write the PreAP curriculum guide so I wanted to make sure it was a good lesson.  Well, at least a better lesson than what was offered (which was “graph these 5 equations at the same time on the TI 83, come up with generalizations like ‘if |a|<|b| then r can be negative which forms an inner loop’, then forget what you just generalized and use a t-table to graph these.”)

I knew I wanted to use desmos and I knew I wanted them to make generalizations.  My coteacher and I decided that my first draft was asking a bit too much (like the original lesson did), then I found this great worksheet and decided to base my lesson on that, using Desmos instead of the graphing calculator because DUH.

Here is the word file and the desmos file. The students worked in pairs in the computer lab and except for mention to a few of them that you can use the sliders instead of typing in each equation (I left off “like it says in the directions” so I wouldn’t have to put a quarter in the sarcasm jar), I didn’t say anything to guide them.  About 3/4 of them got through during the 50 minute class, with the rest having to finish up the last row for homework.

Now this class hates discovery learning with the force of a thousand suns.  And they loved this! (or at least they told me they did and they were engaged and I didn’t hear any of the normal “I can’t learn unless you teach me.”)  AND they made a lot of the discoveries I wanted them to, some of them more formalized than others, but all of them on the right path.  Plus a few of them even had time at the end and chose to spend it playing with Desmos.

It also made the next day so much easier because they knew (a) what the graphs should look like (instead of my horrible attempts at drawing) and (b) what some parts of the equation did.  I made this modest notetakermaker for the day:

It took most of the period to discuss and they had a matching worksheet for homework.  Tomorrow we’ll practice graphing by hand and solving systems.

Now I’m sure you’re wondering how this lesson was bagel award-winning.  As I may have previously mentioned, @bronilo16 is the world’s best math department chair who took on a fourth prep of PreAP Precal so I didn’t have 32+ students in 3 of my classes.  I told her I would do all the prep work, but of course, since she’s awesome, we still spend a lot of time hashing out ideas (she’s also the Calculus teacher so she knows what to focus on and she has a ton of ways to make ideas easier–the notetakermaker was just a prettified version of her notes).  So I’m running late to class Friday morning because of a meeting and she’s waiting in the hall with a bagel for me because of how awesome these lessons went!

That + a thank you letter in the mail from a current student + another student letting me know that I’m “quirky in a good way” because I “probably don’t hear that enough” = one awesome week of school!!  Since I spent the first 6 weeks of school crying at least twice a week, this was quite the turnaround week!  Maybe I will be able to make it until May!

But now the pressure is on and the next lesson is complex numbers….. Eek!

## Sunday Summary

Posted on 1 comment

3 things to share

1. Man, don’t you hate it when you figure out a better way to teach something the day after you teach it?  Although trig equations went pretty well this year (well, we’ll see tomorrow on their quiz), I think next year I will structure it differently. Here’s where we started:

Next year, I’m going to start at the end with the calculator/desmos, with a -4π to 4π window to discuss the general form.  Then do some examples with θ between 0 and 2π, then non-calculator examples.  Then on day 2, graph 2θ (or θ/2) on the same graph to discuss getting more/less answers.  If you’d like to modify this for me (doesn’t hurt to ask, right?), or if you want to use it as is, here is the .doc file (with bonus homework at the bottom!)

2. Now, day 2’s note-taker-maker, I’m kind of in love with. (ok, technically this was day 3 because I ended up teaching day 1 slowly)

The only thing I might change is super-reinforcing the ZERO product property rule is not the “ZERO or sometimes 2 or 3 or -7 product property rule” because some of my students are still having issues with that. (btw, Snoops is dancing because it’s already factored.  Also, it is super fun to have a kid ask later, “How can we solve this?” and you reply, “Cute and cuddly, boys!”  File link.

3. I finally figured out about 3 years ago how to make linear programming less painful…get the mechanics of it out the way first!

Let them spend a day finding the feasible region, vertices, and max/mins.  Then the next day you can focus on the finding the equations from those long scary problems and the rest is the same as these notes.  Much less stress than trying to introduce all of it the same day.  File here.

2 good books I’ve been reading

We took a quick weekend trip to Chattanooga last week where I started David Benioff’s City of Thieves and could not put it down!  It’s like a buddy cop movie set in the absurdity of World War II.  It reminded me of Anthony Marra’s Constellation of Vital Phenomona, one of my favorite books of the last few years.

Hector Tobar’s Deep Down Dark is the reason I haven’t gotten anything done today!  How did he make this so compelling when I already know the outcome?  I’m about to go draw a bath and try to finish it tonight; it’s such a page turner!  And don’t worry, he does a really good job making sure the reader doesn’t get lost with all the people in the story (I’m horrible at remembering names in both book and real life.)

I have to write a unit plan about complex, polar, and parametrics using some premade lessons (and adding others as I see fit). The premade lessons are really expecting a lot from our students and I’m not sure if I will get the outcomes desired by using them (unless the desired outcomes are tears and frustration).  But I’m having trouble finding a lot of great stuff that I can easily replace them with (I have a few things thanks to @mrdardy and @crstn85).  So if you have some cool stuff to share, please do so!  The state of Alabama will thank you!

## #mtboschallenge 1-2-3 Sunday Summary

Going to change up my order a bit so I can start with:

One Totally Awesome Way to Deal with Complex Fractions (complete with poor cell phone visuals)

I don’t know where I picked up this method (I actually want to say it was a textbook and not MTBoS, but that doesn’t sound right, so if it was from you that I got this, let me know) but it is a totally awesome way to deal with complex fractions.  Before, I would simplify the top and bottom, then multiply the numerator by the reciprocal of the denominator.

I was going to use arrows to show the two different ways on the same page but I ran out of room.  It’s not that I don’t think you could follow my steps without them.

Maybe to save time, once I simplified the top and bottom, I’d use “outers over inners” to get the next step, while pretending I didn’t know what this Nix the Tricks thing was all about.

Now here is the totally cool, time-saving, wait-til-you-get-to-cal-to-see-how-awesome-this-is, not-a-trick way instead.

Multiply the top and bottom by the LCD of the “tiny denominators.”

WHAT???  How amazing is that?  We just have to choose a “convenient one” and all that middle work is taken care of.  Some more examples:

I need to set up some sort of royalty system where I get 3 cents every time you use this rule and are completely amazed by it (because I still am amazed after using it for a year).  (I also think all those Shark Tank viewings may be going to my head.)

Two Other Cool Things To Share

1. What reminded me about the cool fraction trick was the fact that we were working with trig identities and double/half angle formulas this week.  Here is a lovely handout (.doc file with “running for a cause” font) for trig identities:

The order of the identities works out nicely to do 1-6 as a group, 7-12 as individual practice, then the rest as group practice (and yes, 11 and 19 are the same. My laziness strikes again.)  I also tried to disperse the more difficult ones throughout the worksheet instead of all at the end.

2. Mattie B (@stoodle) sent out this plea this week:

Which led to a discussion of Bowman’s Dead Puppy Theorem, which then led to my sharing of this handout, (.doc file with “running for a cause” font) which has all of the ways to save animals.

You’re welcome, I already test drove this and therefore changed save “yourself” to “your grade” to avoid the awkward “don’t kill yourself” statement.

The arrows at the top are a reference for this fabulous way to think about exponents from Sweeney Math

Shoot, Now I Need to Think of Three Other Things to Talk About

1. I tried to go vertical with my group whiteboards this week by having my Precal classes work on their identities.  It was AMAZING!!  I could do a quick scan to see where everyone was, 98% of the kids were engaged, if we had time at the end of class I could say “to the boards!” and they would pick up where they left off the day before.  Then all the whiteboards fell off the wall overnight.  I’m just thankful the one that did fall off during class wasn’t where any students were sitting.  I put in a request to see if someone would mount them for me (or if I would be allowed to mount them), but haven’t heard back yet.

2. Like many of the other MTBoSers, this has not been a very positive year overall for me.  All the grand plans I had at the start of the year have been squashed and now I’m basically back to doing the same things that I’ve always done.  But now with 20% more stress.

3. Have you tried showme? I was introduced to it by a new teacher at my school, and it is pretty cool.  Unlike some of the other ipad whiteboard apps, this one makes and stores the video for you–no need to find a place to upload it or deal with a password-protected site.  The only thing I wish it had would be the ability to import PDFs, but another teacher had the brilliant idea of taking a screenshot of the PDF and importing the picture.  Clever!